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Core 3 Natural Logarithms

Core 3 Natural Logarithms. Learning Objective: Understand what a natural logarithm is. Logs (1). Laws of logs: log a + log b = log ab Example: log 3 + log 5 = log 15 log a - log b = log ( a/b) Example: log 21 – log 7 = log 3. Logs (2). Laws of logs:

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Core 3 Natural Logarithms

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  1. Core 3 Natural Logarithms Learning Objective: Understand what a natural logarithm is.

  2. Logs (1) Laws of logs: • log a + log b = log ab Example: log 3 + log 5 = log 15 • log a - log b = log (a/b) Example: log 21 – log 7 = log 3

  3. Logs (2) Laws of logs: • log a k = k loga Example: log 35 = 5 log 3

  4. e = 2.718281828459045…... ‘the Base of Natural Logarithms’ x ( 1 + 1 ) It is defined as : lim x X The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant). An effective way to calculate the value of e is to use the following infinite sum: e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ….

  5. Why e ? e is a real number constant that appears in some kinds of mathematics problems. Examples of such problems are those involving growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, and even the study of the distribution of prime numbers. It appears in Stirling's Formula for approximating factorials. It also shows up in calculus quite often, wherever you are dealing with either logarithmic or exponential functions. There is also a connection between e and complex numbers, via Euler's Equation. Also, you’ll soon see, it’s unique properties make it very powerful in calculus

  6. y = ex When x=0 y = e0 = 1 So it goes through (0,1) The x-axis is an asymptote y = ex can, of course, be transformed

  7. Natural Logarithms • Any log to the base e is known as a natural logarithm. • In French this is a logarithme naturel • Which is where ln comes from. • When you see ln (instead of log) • then it’s a natural log

  8. The laws of logs still hold Laws of logs: • ln a + ln b = ln ab Example: ln 2 + ln 8 =ln 16 • ln a - ln b = ln (a/b) Example: ln 42 – ln 6 = ln 7

  9. and for these Laws of logs: • ln a k = k lna Example: ln 35 = 5 ln 3

  10. Natural Logarithms eg1 Remember: log 10 = 1 Similarly, ln e = 1 • Solve: ex = 7 • can use trial and improvement • x=1.95 • or use the ‘Laws of Logs’ ln ex = ln 7 x ln e= ln 7 x = ln 7 x = 1.9459 Which is quicker and more accurate

  11. Natural Logarithms eg2 Remember: log 10 = 1 Similarly, ln e = 1 • Solve: 2 e3x = 7 • use the ‘Laws of Logs’ e3x = 7/2 = 3.5 3x ln e= ln 7/2 3x = ln 7/2 3x = 1.2528 x = 0.418

  12. Natural Logarithms eg3 Remember: log 10 = 1 Similarly, ln e = 1 • Solve: e3x+2 - 1= 7 • use the ‘Laws of Logs’ e3x+2 = 7 + 1 = 8 (3x+2) ln e= ln 8 3x+2 = ln 8 3x+2 = 2.079 3x = 2.079 – 2 = 0.079 x = 0.079/3 = 0.026

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